For the linear programing problem, Minimize(Z) = 60x + 30y subject to: 2x - y ≥ -5; 3x + y ≥ 3; 2x - 3y ≤ 12; x, y ≥ 0 the optimal value of z is

Given three vertices of parallelogram ABCD:
A(3, 2)
B(1, -1)
C(2, 1)

The area of a parallelogram can be found using any three consecutive vertices by calculating the cross product of two adjacent sides.

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Vector AB represents the side from A to B:

Vector AB = B - A

Vector AB = (1 - 3, -1 - 2)

Vector AB = (-2, -3)

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Vector AC represents the side from A to C:

Vector AC = C - A

Vector AC = (2 - 3, 1 - 2)

Vector AC = (-1, -1)

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For a parallelogram with adjacent sides represented by vectors (x₁, y₁) and (x₂, y₂):

Area = $|x₁ \times y₂ - x₂ \times y₁|$∣x1​×y2​−x2​×y1​∣

Substituting AB = (-2, -3) and AC = (-1, -1):

Area = $|(-2) \times (-1) - (-1) \times (-3)|$∣(−2)×(−1)−(−1)×(−3)∣

Area = $|2 - 3|$∣2−3∣

Area = $|-1|$∣−1∣

Area = $1$1

Therefore, the area of parallelogram ABCD is 1 square unit.